Soflaques finishing sauce and mustard sauce

I am not sure and would also like input from the chefs.  I have made Soflaques finishing sauce put in a used vinger bottle and left it on the counter for a week using it as I reheat PP.  Should I refrig it?  All BBQ sauces I make I vac seal in pint size jars and seems to keep a  couple months in the refrigerator.  They don't last much longer than that.

Stan